Thanks Tom & Chris.
Fortunately I am not the type who believes “the science is settled” or in this case “the etymology is settled”. Alternate theories should always be taken into account.
I believe Mrs. Stewart is correct that the proper name for the knitted cord is Lilly Yarn 「リリー・ヤーン」.
But that name also seems to be derived from the Lilly flower. " Back in 2009 I asked my wife what リリアン meant. She said it was lilly yarn, …and called the resulting yarn lilly yarn because of the knitting spool was in the shape of a lilly." Also seems to confirm the name derives from the lilly flower.
At first I was not sure there will be any need to change the name. But on further expiration, it appears that the name lilian リリアン really refers to the tool, the knitting spool itself. And is not the proper name for the knitted string made using the tool. Which is lilly yarn 「リリー・ヤーン」 or string, cord, what ever alias name. リリヤン糸.
However, the string attached to our rod tips, ロッド穂先 (roddo hosaki). Mostly have the name リリアン, in the product names. Such as リリアントップ (Ririantoppu) ririan top, トップ リリアン (toppu ririan), リリアン糸 (Ririan-ito) ririan string, and 回転式リリアントップ , (Kaiten-shiki ririantoppu) rotary lillian top. And even リリアンタイプ (ririantaipu ) lillian type.
リリアン is the name they put on the packages, for the replacement parts. Often omitting to add 糸 to the name.
オリジナル回転式リリアントップ ゴールド , Original rotating lilian top gold. (sorry the picture was too big to embed)
Original rotating lilian top gold
The リリヤン - Wikipedia web-page states that:
the whole seller in 1923 sold the knitting spool / knitting ring tool, under the trade name of リリヤン (Ririyan).
「リリヤーン」の名で。Under the name of “Ririyan”.
Using（ユリ・印・糸）Lilly (flower) Mark Thread.
リリヤン (ririan)（Lily-yarn）とは、[what is ririan?] 細く編み込んだ手芸用のひものこと。Handcraft thin knitted string [ひも, himo, string]
If you do an internet search for ユリ (yuri) you find Lilly flowers.
And on the リリー - Wikipedia web page. Among many other meanings:
リリー (Rirī ). Is another or alias name for ユリ (yuri). The Lilly flower. ユリの花 (Yuri no hana).
I suspect there is a bit of word play involved in the name lillian too.
Perhaps similar to the word play with the word テンカラ, tenkara, as 十カラー ( Jū karā ) 10 colors. vs the kanji version of 十色 (Toiro) or to keep the phonics from changing 十・色 (Jū-iro) Ten-color.
If you enter リリヤン into the Japanese input window in google translate, and click the speaker icon, you can hear an electronic pronunciation of the word. And clicking on the English translation of Lilyan, an alternate translation is - Spool knitting. Of course google translate is often “out to lunch” We both dine together frequently, 
It’s easy to see how the name knitting spool came about by people making their own knitting ring from a small wooden thread spool.
How to make your own spool knitter
