okay, I think the "lallygag" spelling has gotten the sticky end of the stick here, which only promotes some non-existant connection to "lollipop". so here is the history according to OED2:
lallygag - v. U.S. slang.

('læligæg) Also lollygag. [Origin unknown.]
intr. To fool around; to ‘neck’; to dawdle, to dally. Also as n., fooling around. Hence "lallygagging vbl. n. and ppl. a.
1862 Harper's Mag. Aug. 324/1 Mr. Biggs paused and turned the flesh of the succulent lobster over with his finger. The gentleman inside addressed him: ‘+Try er lobstaw, bossy?’ ‘Ain't got no money,’ said Mr. Biggs, still fingering the morsels. ‘Oh, come now, none o' that ere lallygag,’ responded the gentleman. 1868 Northern Vindicator (Estherville, Iowa) 30 Dec., The lascivious lolly-gagging lumps of licentiousness who disgrace the common decencies of life by their love-sick fawnings at our public dances. 1869 Tidal Wave (Silver City, Idaho) 15 Jan. 3/2 They are too pious to encourage dicing, and the feature of their entertainments may be what the boys call ‘lally-gagging’. 1870 Northern Vindicator (Estherville, Iowa) 19 Feb., The weather once more is ‘salubrious’ and balmy, and indicates that winter will not lollygag in the lap of spring. 1880 E. L. Wheeler Boss Bob, King of Bootblacks vii. 9/1, I kin get lots o' jobs, if I'd take my pay in friendship an' all sech lollygag. 1910 Sat. Even. Post 30 July 19/1 Frank lally-gagged through his first term and came back for the second. 1927 D. Runyon Trials & Other Tribulations (1947) 112 When your correspondent was a ‘necker’ of no mean standing back in the dim and misty past, they called it ‘lally-gagging’. 1949 Jrnl. Amer. Folk-Lore Jan.—Mar. 63 ‘Lally-gaggin’ was Grandmother's word for love-making. 1965 ‘E. Queen’ Fourth Side of Triangle i. 2 Lallygagging around under the awning away from the gassy streets. 1969 S. Greenlee Spook who sat by Door xi. 95 We lolly-gag, maybe turn on, or cook up some soul food. 1971 D. Bagley Freedom Trap iii. 20, I said there was to be no lally-gagging around with the staff, Rearden; you just stick to doing your job. 1973 Springfield (Mass.) Union 25 Sept. 14/1 The Dow Jones average of 30 industrials, which lollygagged most of the day, gained strongly in afternoon trading.


(citations don't lie)

winter will not lollygag in the lap of spring?!