I have this hazy memory from seventh grade English class of there
being a mythical river that was supposed to span a continent--I'm not
even sure which--from coast to coast.  I thought it was North America,
and I thought the river sounded like "Anais," (or uh-nye-is,
phoentically) but I can't find anything about it.  It's been quietly
nagging me at various times over the last decade.

Bottom line: Am I making this up?  If not, what's the name of the river?

Your memory isn't playing tricks on you. It was once thought that
there was a waterway that spanned the North American continent, and
the name of the mythical waterway does sound like 'Anais.'

What you are remembering is the Strait (or Straits) of Anian
(sometimes spelled 'Anián or 'Anion').

"The Strait of Anián was the 16th century Spanish name for the
Northwest Passage that supposedly connected the Pacific Ocean and
Atlantic Ocean in the temperate or tropic regions of North America. It
was based on the idea that North America was not a continent but a
large archipelago. Such a strait does not in fact, exist, although the
Northwest Passage itself was discovered in the 20th century through
the Canadian Arctic islands."

Wikiverse: Strait of Anian
http://strait-of-anian.wikiverse.org/ 

"In 1592 Juan de Fuca sailed up the west coast of North America
looking for the Straits of Anian, that mythical waterway thought to be
between the Atlantic and the Pacific; otherwise known as the Northwest
Passage. Between 47 and 48 degrees north of the equator he found the
mouth of a large inlet which he took to be a likely candidate."

InterIsland: The Spanish Influence in the Pacific Northwest
http://www.interisland.net/abi/sh.html

"During the early exploration of the North American continent it was
rumoured that there was a sea route through the continent below the
polar ice cap and it was given the name 'the Strait of Anion.' During
the latter part of the 18th and early part of the 19th centuries,
until it was proven that the Strait of Anion did not exist, European
ships explored the west coast of British Columbia looking for the
mythical passage."

Fly Anglers Online: Great Rivers
http://www.flyanglersonline.com/features/greatrivers/dean/

"To that long-continued, that reluctantly-abandoned faith in the
existence of the Strait of Anian, or the northwest passage, is to be
attributed those voyages which mark the early exploration of the coast
of Northwest America. Kindred with the thought which accepted as
assured the existence of that mythical strait, indeed, intensifying
the mystery and co-operating to render those coasts more inviting to
adventure, were fabulous narratives of pretended voyages and
discoveries, which for centuries were credited."

USGenNet: History of the Pacific Northwest 
http://www.usgennet.org/usa/or/county/union1/1889vol1/1889volumeIpage1-10.htm

Google search strategy:

Google Web Search: "strait of anián OR anian OR anion"
://www.google.com/search?hl=en&ie=UTF-8&q=%22strait+of+ani%C3%A1n+OR+anian+OR+anion%22

I hope this is helpful! If I have misidentified your waterway, or if
anything is unclear or incomplete, please request clarification; I'll
be glad to offer further assistance before you rate my answer.

Best regards,
pinkfreud

This is exactly what I was trying to remember.  In addition to the
name of the waterway, the researcher provided historical context for
it.  I am completely satisfied.

Thank you for the five-star rating and the nice tip!

~pinkfreud